Author:
Wade A. Tisthammer
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Date Posted: 05/27/04 3:04pm
In reply to:
Duane
's message, "Wade was right (sort of)" on 05/25/04 11:03am
>Wade:
>
>That last thread got too cluttered, so I started a new
>one.
>
>After a little more thought, I realized the following
>about your first premise (that there is a 1 to 1
>correspondence of days to years).
I should again caution about the existence of the revised argument (e.g. see this post). I no longer believe the original argument works.
>You defined time as a set in which each element
>corresponds to 365 distinct non-self elements that are
>(arbitrarily) less than the element in question,
>within that same set.
Not really. Just that for every day there is a year. It's actually two sets, though taken from the same set (time).
>Take the empty set - now add one element. There's a
>day, so he spends 365 more days writing about it. At
>the end of that year, as we've added individual
>elements to the set, we end up with 366 individual
>elements.
>
>Now we've satisfied your initial condition, that day 0
>has a one to one correspondence to year 0 (days
>1-366). But, in doing so, we've added 365 more days
>that also must have, each 365 distinct, greater
>elements to correspond to each, so your condition (1
>to 1 correspondence) isn't true.
Except that this doesn't accurately represent what I was talking about. In an infinite past we have a one-to-one correspondence. Think I'm wrong? Name a year that doesn't have a corresponding day. (Remember the scheme was to count backwards; year 1 is last year, year 2 is two years ago etc.)
>The 1 to 1 correspondence is not true for any finite
>subset of natural numbers, and I claim we may state
>that it is untrue for an infinite set of natural
>numbers.
Then again, name a year in the infinite past that doesn't have a day. In an infinite past, there is the function Y(d) = d, where Y is the year for the given day d. What we have here is the identity function; there will always be a year for a given day no matter what number d you input.
>This is so because it is not true for the basis case
>(1 element in the set), nor for the first recursive
>step (when we add 365 days), nor the second, etc.,
>etc. This satisfies the requirements for proof by
>induction, and shows that the property (i.e., the 1 to
>1 correspondence of days to years) is not true, ever.
For any finite set that certainly is true, for any real number of cardinality. The problem is that infinity isn't a real number. You can't name me even one day that doesn't have a year to match up with.
You see with mathematical induction, we encounter a similar problem with Count Int. No matter how far you get, you never reach infinity. So we can't possibly expect the mathematical induction to apply to infinity, since no matter how many recursive steps we do, infinity is never reached. (Using mathematical induction, we could similarly prove that infinity can never be formed via successive finite addition.)
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